On the orbit closure containment problem and slice rank of tensors

We consider the orbit closure containment problem, which, for a given vector and a group orbit, asks if the vector is contained in the closure of the group orbit. Recently, many algorithmic problems related to orbit closures have proved to be quite useful in giving polynomial time algorithms for special cases of the polynomial identity testing problem and several non-convex optimization problems. Answering a question posed by Wigderson, we show that the algorithmic problem corresponding to the orbit closure containment problem is NP-hard. We show this by establishing a computational equivalence between the solvability of homogeneous quadratic equations and a homogeneous version of the matrix completion problem, while showing that the latter is an instance of the orbit closure containment problem. Secondly, we consider the notion of slice rank of tensors, which was recently introduced by Tao, and has subsequently been used for breakthroughs in several combinatorial problems like capsets, sunflower free sets, tri-colored sum-free sets, and progression-free sets. We show that the corresponding algorithmic problem, which can also be phrased as a problem about union of orbit closures, is also NP-hard, hence answering an open question by Bürgisser, Garg, Oliveira, Walter, and Wigderson. We show this by using a connection between the slice rank and the size of a minimum vertex cover of a hypergraph revealed by Tao and Sawin

Twenty points in p3

Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in P3 over an algebraically closed field of characteristic 0. As a consequence, we show that a general set of d points is glicci (that is, in the Gorenstein linkage class of a complete intersection) if d ≤33 or d = 37, 38. Computer algebra plays an essential role in the proof. The case of 20 points had been an outstanding problem in the area for a dozen years [8]. For Rob Lazarsfeld on the occasion of his 60th birthda

Rank two aCM bundles on general determinantal quartic surfaces in \p3

Let F\subseteq\p 3 be a smooth determinantal quartic surface which is general in the N\"other--Lefschetz sense. In the present paper we give a complete classification of locally free sheaves \cE of rank $2$ on $F$ such that h^1\big(F,\cE(th)\big)=0 for t\in\bZ